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We consider subspaces of Morrey spaces defined in terms of various vanishingproperties of functions. Such subspaces were recently used to describe theclosure of $C_0^\infty(\mathbb{R}^n)$ in Morrey norm. We show that thesesubspaces are invariant with respect to some classical operators of harmonicanalysis, such as the Hardy-Littlewood maximal operator, singular typeoperators and Hardy operators. We also show that the vanishing propertiesdefining those subspaces are preserved under the action of Riesz potentialoperators and fractional maximal operators.

On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators